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DESIGN AND CONSTRUCTION 
OF HYDRAULIC FLUME AND BACKWATER EFFECTS 
OF SEMI- CIRCULAR CONSTRICTIONS 
IN A SMOOTH RECTANGULAR CHANNEL 




Progress Report- Ho* 2 



ID; K« B* V.'oodSj, Director 

Joint Highway Research Project 

FRGMs K» lit Mchael s Assistant Director 
Joint Highway iteseareh Project 



January 21 9 I960 



Ellas 9-S-2 
Project: C«-36-»62I 



Attached is a progress report entitle^ "Design and Construction 
of Kydraulic Fluss and Backwater Bffeots of SeHii«*Cireular Constrictions 
in a SiEOoth Rectangular Channel"* This report has hs&n prepared by 
Mr« He J» 0wsn 5 grsdu&ta research assistant oa cm 1 staff 9 un<fer the 
direction cf J* W, Delleur* Mr* Osjan also utilised the report as his 
thesis in psrtial fuifiilnsnt for the r®o;aireiisnt of the M»8»C*E* 
degree* 

The rsaterial reported in this report is a suuseary of the programs 
■that has occurred on the Hydraulics of Arch Bridges Project ishich is 
being conducted in cooperation t&th the Indiana State Highway Bepajpfessnt 
and the U» S« Bureau, of Public Roads* Copies of this report *411 a2s© 
be distributed to the State Highway Department and the Bareaa of Public 
Roads for thsir review and coKsneats* 

t The report is presented to the Board for the record* 

Respectfully subaittedg, 



HZcisksse 

Attaehnsnt 



cc: F. 



Ashbaechep 



J e R* Cooper 

W» L« Bolsh 

¥. H« Gcets 

P. P. Ba-gsy 

G. A. Ea&l&ns (M» B, Scott) 

G» A. Leonards 



H» L. Michael, Secretary 



J. F* 2feX&ughli» 

R. B» Miles 

R> 3, Kills 

C» S* Vogelgosang 

J. L. Waling 

J* E. Wilson 

F.» J. Yodsy 



Progress Report Ko» 2 



Design and Construction of Hydraulic rTcaae 
and 
Backwater Effects of Sead-Circular Constrictions in 
A Smooth Eectangular Chamol 



Graduate Assistant 



Joint Highly Research Project 
Project Bo, C~36«42B 

nis No« 9«&*a 



Pardw University 
Lafayette, Indians 

.January 21, I960 



A J I'd I U i , hti. IG-M.CJN ! 3 

The author wishes to acknowledge and thank Dr. J. W. 
Delleur whose guidence and help was invaluable. Appreciation 
must also be expressed by the author to his wife .Pat. With- 
out her help, the preparation of the manuscript would have 
been difficult. 



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TABLE OF CONTENTS 

Page 

LIST OF ILLUSTRATIONS v 

LIST OF TABLES vii 

ABSTRACT viii 

INTRODUCTION 1 

REVIEW OF THE LITERATURE 2 

DESIGN AND CONSTRUCTION OF THE APPARATUS 5 

Testing Flume 5 

Preliminary Computations 5 

Length of Backwater Profile 7 

Design Procedure 10 

Dead Load 12 

Live Load 12 

Tail Gate 14 

Forebay 14 

Slope Control Mechanism 15 

Construction 17 

PREPARATION FOR TESTING 18 

Slope Calibration 18 

Water Supply and Measurement System 19 

Venturi Calibration 21 

MODEL TESTING 22 

Model Construction 22 

Free Surface Measurement 23 

Uniform Flow Calibration 24 

Testing 25 

Test Results 27 

ARY OF RESULTS AND CONCLUSIONS 32 

Flume Design 32 

Testing 32 



IV 

BIBLIOGRAPHY^ ' . .' 34 

APPENDIX A NOTATIONS 36 

APPENDIX B EQUATIONS OP PLOW 39 

APPJSMDIX J jj'IGUREo AND TABLES 41 



LIST OF ILLUSTRATIONS 
Figure Title Page 

1. Idealized Stream cross section 41 

2. Conveyance versus Depth Curve 42 

3. Length of Regain 43 

4. Flume Construction 44 

5. Flume and Models 45 

6. Tail gate Construction 45 

7. Plan View of Jacks and Gearing 46 

8. Jack Detail 47 

9. Slope Jalibration Curve For Hydraulic Testing 

Flume 48 

10. Apparatus Arrangement 49 

11. Calibration Curve for 3 Inch Venturi 50 

12. Calibration Curve for 6 Inch Venturi 51 

13. Calibration Curve for 6 Inch Venturi 52 

14. Flow in Rectangular Channels with Semi- 
circular Constrictions-Comparison of two 

and Three Dimensional Cases 53 

15. Model Construction 5^ 

16. Instrument Carriage 55 

17. Model in Place 55 

18. Model in Place 56 

19. Model in Place 56 

20. Normal Depth Versus Slope for Testing Flume ... 57 

21. Slope Versus Gate iieigiith for Testing Flume ... 57 



VI 



22. Superelevation Versus Kineticity . . . * 53 

23. Discharge Coefficient Versus Kineticity 59 

24. Friction Factor Versus Reynolds iiumber 60 

25. Friction Factor Versus Reynolds Number 61 

26. Backwater Ratio Versus Contraction Ratio 62 

27. Definition Sketch 53 



Vll 



LIST OP TABLES 

Table Page 

1. Observed and Calculated Data 64 



Vlll 



ABSTRACT 

a. James Owen, lw.S.C.E. Purdue University, January 
I960. "Design and construction of Hydraulic Flume and 
.backwater Effects of semi-Circular Jonstrictions in a 
smooth Rectangular Ohannel". Major Professor: Dr. J. W. 
Delleur. 

The purpose of this study was to investigate the 
hydraulics of semi-circular constrictions in smooth rect- 
angular channels. To this end a large part of this work 
consisted of the design and construction of a hydraulic 
flume 54 feet long and 5 feet wide. This study is part 
of a general program sponsored by the State Highway De- 
partment of Indiana and the U.S. Bureau of Public Roads 
at Purdue university on the hydraulics of River Flow Under 
Arch bridges. 



INTRODUCTION 

This project was initiated in the Hydraulics laboratory 
of Purdue University by the Indiana State Highway Departi i 
in cooperation with the U.S. Bureau of Public Roads, to study 
the backwater and regain phenomena associated with arched 
constrictions such as sre presented by arch bridges. The 
problem was an- roached by both theoretical analysis and model 
study. The preliminary analysis of the general problem was 
made by Mr. S. T. Husain^ ' . Mr. A. A. Sooky^ ' derived 
exact and approximate equations for the two dimensional sharp 
edged constrictions, and carried out a preliminary testing 
program in a small flume. The present study reports on the 
design and construction of the flume for the main testing 
program. The design was begun in July, 1958 and construction 
was started in March, 1959. The flume and its appurtenances 
were constructed and operative in August, 1959. Also in- 
cluded is the testing of two dimensional semi-circular con- 
strictions carried out from September 1959 to November 1959. 



Superscripts refer to bibliography at end of text. 



ttdvlliiW Ui)' THE LITS^ATUftE 
i.ane v , in 1915 and 1916, performed several exper- 
iments on contracted openings of various tyres. The full 
series of tests was not completed but data was collected 
and evaluated on the following: 

1. Sharp edged vertical contractions 

2. A rounded edge vertical contraction 

3. A short flume with rounded entrance 

4-. a short flume with sharp corner entrance 

5. An expanding flume 

The paper describes the apparatus and proceiure util- 
ized as well as the results obtained. Lane found that the 
JVeisbach formula { Q = Cp b )/2j~ (% H '* + i/ H% ] 
where H is the total drop of the water in passing through 
the contraction including the change in the velocity head 
and y is the average downstream depth. ) wis most nearly in 
agreement with the experimental results obtained in the case 
of the sharp edged contractions. 

Kindsvater aril darter^ ' attempted to correlate labor- 
atory results with field data obtained by the USGS. A gen- 
eral equation was developed for discharge through a con- 
tracted opening: Q.-Qphy ^E^h^T^fj where y 

* See list of symbols in Appendix A 



is the downstream depth. This expression contained a dis- 
charge coefficient which was determined for several cases. 
The base curves of the discharge coefficient in terms of 
the contraction ratios and length of the constriction were 
presented for a standard condition of square abutment and 
Froude number of 0.5. The base curves are supplemented by 
auxiliary curves modifying the coefficient due to: 

1. Frou.de number different from standard 

2. Upstream corner rounding 

3. Eccentricity 

4. Skew 

5. Channel contraction 

6. Chamfers of the abutments 

7. oide depths 

8. Abutment side slopes 

9. Submergence 

10. Bridge piers and piles 
Tracy and barter ' made a laboratory study in 195^ of 

the flow through contractions. Tiie data was used to develop 

* 
a set of base curves relating the backwater ratio (h,/ & h) 

and the contraction ratio jvi for different values of roughness 
These base curves were obtained using vertical-faced con- 
strictions v/ith square edged abutments, ani a Froude number 
of 0.5- Auxiliary graphs were presented to modify the result 
according to the geometry of the constriction if other than 
the basic geometry. 

Vallentine ' used vertical sharp edged constriction 
plates placed normal to the flow. The discharge through the 



constriction was related to the upstream' depth by an equation 
containing an experimental discharge coefficient: Q} = 
dn b vZq *j ^ • ^ke variation of this discharge co- 
efficient with the Froude number of the unconstricted flow 

was given for several values of the contraction ratio. 

(7) 

jNagler did considerable experimental work with three 

dimensional models of bridge piers. Although he worked at 
an extremely small scale, several curves are presented re- 
lating the backwater to tne geometry of the pier. 

Husain carried out two ani three dimensional tests of 
archei openings on a small scale in preparation for a larger 
study of which the work reported here is a part. The prob- 
lem was also approached by dimensional analysis. General 
profiles of the backwater curves were obtained ani recom- 
mendations were made concerning the design of the future 

equipment . 

( 2 ) 
oooky^ ' carried on the small scale testing program be- 
gun by Husain and included two and three dimensional models 
with and without channel roughness. For the two dimensional 
case, equations relating the discharge to the geometry of 
the opening ana the backwater height were developed. Curves 
were obtained from experimental data which relate the coef- 
ficient of discharge and the backwater height to the Froude 
number of the unconstricted flow and the contraction ratio. 
These equations are presented in Appendix B. 



DESIGN AND CONSTRUCTION OF THE APPARATUS 

Testing; Flume 

Preliminary Computations 
Before the design of the flume itself could proceed, 
it was necessary to determine whether the backwater and 
regain phenomena could be represented to a convenient and 
easily measurable scale in the space available in the hy- 
draulics laboratory. 

Several sets of arch bridge plans provided by the 
Indiana State Highway Department were analyzed for the 
values of backwater. The theory of varied flow and the 

equations and tables presented by Bakhmeteff ^ J and by the 

(a) 

U.S. Bureau of Public Roads w/ were used. It was only pos- 
sible to make an approximate calculation of the backwater 
because of the unknown effect D f an arch-type constriction. 
A sample calculation is presented for bridge S79 (Clay 
County , Ind i -ana ) . 
Assumption 

1) Velocity through the bridge constriction 

= 6 feet/second. This value was suggested 
by Mr. J. I. Perry, Chief Engineer, Indiana 
State Flood Control and Water Resources 
Commission, as an average flood flow vel- 
ocity through typical bridges in Indiana. 



From bridge plan 

.Vaterway area through the bridge utilized by the 

2 
stream = 125 feet 

Slope of the stream bed = 0.0019 feet/foot 

Flow = 750 cfs 

The total waterway area under the arched constriction 

of the bridge was approximated by a trapezoid. (See figure 

1 appendix G) The unconstricted area of the waterway up- 

2 
stream of the bridge was found to be 300 feet . 

In order to find the percentage of constriction of the 
stream it was necessary to proportion the flow within the 
subsections of the original area. 

The tabulation for the prooortionment is shown below. 



Sub 


n 


1.486/n 


A 


P 




3 


Section 






feet 2 


feet 




feet 


1 


.05 


29.3 


45 


12 




3.75 


2 


.05 


29.8 


210 


30 




7 


3 


.05 


29.8 


^5 

A = 300 


12 




3.75 


Sub 


R 2/3 


AR 2 ' 5 


K 




Q 




Section 






cfs 




cfs 




1 


2.41 


108.5 


3,240 




86 




2 


3.65 


776.0 


22,800 




578 




3 


2.41 


108.5 


3,240 




86 





K = 28,280 Q = 750 
The prooortionment was r nade on the basis of conveyance. 
The interface of sub sections was considered as a boundary. 



7 

p 

The total water-way area through the bridge was 143.5 feet 

located entirely in sub section 2. The loss in capacity 

based on relative conveyance was computed as follows. 

2 
Area deducted section 2 = 66.5 feet 

(see figure 1 aopendix C) 

2 
Area total section 2 = 210.5 feet 

Flow deducted section 2 = 

(66.5/210) x 578 = 182 cfs 
Flow deducted section 1 = 86 cfs 
Flow deducted section 3 = 86 cfs 
Total flow deducted = 35-4- cfs 
The loss in capacity represented by this constriction 
is 554- cfs. The contraction ratio M then is (354/750) x 100 
or 47/y. 

The following computations are based on data taken 
from the U.S. Bureau of Public Roads report. ' 

Assuming the geometry of a 30° wing wall normal cross- 
ing, the backwater coefficient K, = 1.075 for a contraction 
ratio M = 47% 

V 2 /2g = 36/64.4 = .56 feet 
h-j^ = 1.075 (.56) = 0.602 feet. This is the maximum back- 
water superelevation. 

Length of Backwater Profile. The backwater profile was 
comouted using Bakhmetef f ' s^ ' function y \V; • The compu- 
tations made on this basis neglect the regain of kinetic 
energy in an expanding stream. In order to use the tables 
of 0\?7) , the hydraulic exponent n characteristic of the 
channel upstream of the bridge was computed by use of the 



equation n = 2 cot oC . (. <tC is shown in figure 2 ap- 
pendix C) 



y = 3 feet y 



1=7 feet 



A = 90 + 18 = 108 feet 2 A x = 300 feet 2 

p = 30 + 12 = 42 feet 2 p, = 44 feet 



1 

1 
R - 108/42 =2.57 feet R ] _ = 300/44 = 6.82 feet 

«J = 1.486/. 05 ^2.57; 1/6 - 34.8 C 1 = 1.486/. 05 x (682) 1/6 = 41.0 

K = 108 x 34.8 x 1.6 = 6000 iL ± = 300 x 41 x 2.61 = 52,100 

Log K = 3-78 Log K x = 4.506 

Log y = 0.478 Log y ± = 0.846 

n = 2(4.506 - 3.78/0.436 - .478) = 2(. 726/. 368; = 3-95 

y 7] *0 i L 

feet feet feet 
7.602 1.086 0.660 

7.500 1.070 0.599 .061 628 628 

7. 400 1.058 0.544 .055 565 1193 

7.300 1.041 0.446 .098 1010 2203 

7.200 1.030 0.362 .084 865 . 3068 

7.100 1.013 0.137 .255 2320 5388 

7.050 1.00S 0.004 .133 1370 6758 

7.020 1.003 0.264 .278 2860 9618 

Original normal depth = 7.00 feet 

backwater = 0.60 feet 

Depth at maximum 

backwater = 7-60 feet 

Superelevation 

decrease = 0.58 feet 



This reach then includes 0.58/0.60 x 100 or 96% of 
the superelevation. 

Complete information is lacking on the length of the 
regain curve. 

It may be assumed that the angle of expansion of the 
stream downstream of the contraction may be approximated by 
the divergence angle of a submerged jet, and that the re- 
gain curve will be complete when the expansion has reached 
the full width of the flume. Albertson^ ' found a diver- 
gence angle on each side of the C3nterline of a free jet of 
11 to 14 degrees. Henry ^ ' found the free boundary to di- 
verge at approximately 7 degrees for the flow from a sub- 
merged sluice gate. An angle of 5 is estimated with this 
particular bridge in order to obtain an approximation by 
excess of the regain curve as follows. (See figure 3, ap- 
pendix C). 

Clear soan of bridge at spring line = 30 feet. 

burface width of arch at maximum high water = 11 feet. 

Average width of opening = ( 30 + ll)/2 = 20.5 feet. 

Stream width = 46 feet 

The regain to normal depth should therefore be complete 
in a length of 146 feet measured from the downstream side of 
the constriction. This approximate computation indicates 
that the total length of the backwater curve reach (within 
0.02 feet of the normal depth) and the estimated regain 
curve plus the bridge is 9618 + 146 + 48 = 9312 feet. It 
is now desired to find the required length of flume to rep- 
resent to scale the totality of the regain curve and a 



10 

reasonable portion of the backwater curve. 

The water flow available in the hydraulic laboratory 
was 2100 Gpm. 

2100 Gpm = 2100/449 =4.7 cfs. 

5/2 
bince <^m/Qp = L the required scale ratio is found 

to be 4.7/750 = L 5/2 = 0.00627- 

Lr = C-00627J 2/5 = .155 

or Lr = 1:7.4 

The nearest convenient scale is 1:10. For this ratio, 
the superelevation = .06 feet = .72 inches. The usable 
length of the flume in the proposed soace would equal 60 
feet. For this ratio, 600 feet of the prototype stream 
could be represented. This would include the totality of 
the regain curve, the bridge model and some 400 feet of the 
prototype backwater curve. 

From this and similar computations as well as small 
scale tests, ' it apoeared that a flume utilizing all of the 
easily available SDace in bhe laboratory that is a flume length 
of 64 feet but capable of extension would be satisfactory. 
The width of the flume w?s fixed at 5 feet. This was based 
on a consideration of the scale ratios and the space available. 

The cross section of the flume was to be rectangular 
since this configuration lent itself well to both ease of 
construction and adaption. 

Design Procedure 

In order to test the flows under varying slope con- 
ditions, the flume was to be tiltable about one end. 



11 

Several methods are available to achieve slope control 
including: 

1. Hydraulic Jacks 

2. Screw Jacks 
J. wedges 

4. Sables and vV inches 

bcrew Jacks were selected because of accuracy and ease 
of control, as well as permanence and appearance. 

At the time the preliminary design wis made, only an 
estimate could be made of the final weight of the flume and 
the water contained therein. 

in order to keep the deflections due to bhe variable 
weight of water within the same order of magnitude of the 
smallest readings of the point gage for depth measurement, 
0.1 mm, the flume bottom was designed of 1/4 inch steel plate 
supported at 2 foot intervals on channels. The channels in 
turn were to be supoorted by two or more main beams riding 
on the jacks. The sile plates were designed of 1/4 inch 
steel plate supported by vertical angles resting on the 
channel members. A longitudinal horizontal angle mounted on 
the vertical angles served as a support for the guide rails. 
The guide rails, which serve as a reference plane from which 
measurements are based, were to be nolished stainless steel 
to minimize corrosion and scale. 

The preliminary design was based on a possible water 
depth of 2. feet. In the immediately proposed tests this 
will allow a freeboard of approximately 1 foot. However, de- 
flection will be within the set limits for a loading of 



12 

2 feet of water which may 0C3ur at a later date. 
A portion of the first design is presented. 
Dead Load 
1/4 inch plate 
2 side plates 64 f set long x 2 faet wide 

@ 20.4 lb/foot = 2620 lb. 

1 bottom plate 64 feet long x 5 feet wide 

© 51.0 lb/foot = 3260 lb. 

5880 lb. 
Main Beam (18 I 54.7) 

2 x 64 feet x 55 lb/feet = 7040 lbs. 
Channels 

53 x 5 feet x 8.2 lb/feet = 1360 lbs. 

14,280 lbs. 
Extras @ 30% 4,300 lbs. 

total 18,580 lbs. 
For design purposes this is an average load of 

18,580 lbs/64 feet = 290 lbs/foot. 
Live Load. At the maximum depth of 1 feet the volume 
of water contained is 10 feet per foot of length. This 
imposes a load of 10 feet^ x 62.4 lbs/foot^ = 624 lbs/foot. 
The total load per foot then is 290 pounds/foot + 624 lbs/foot 
= 914 lbs/foot. 

The distance between supports was set as 20 feet. Since 
the exact nature of the main beam connections was unicnown, the 
solution was made based on a simply supported condition. 

Two alternatives were presented. The first was to use 
beams whose stiffness would make any deflections negligible 



13 

and the second was to use lighter beams and correct for t 3 
deflections by adjusting screws. The first alternative was 
chosen as the second would necessitate adjustment after each 
change in conditions such as slope, or water iepth. 

The beam first selected was an 13 I 5^.7 which gave a 
calculated deflection of 0.00225 feet under the design loading. 
Contacts with the fabricator and erector were made at a later 
date and it was found that a 20 I 65.4- would be available 
at a cost less than that of the lighter beam. The use of 
the heavier beam w:s accepted and the design oroceeded based 
on this beam. The deflection due to the variable water 
weight was approximately 0.002 feet for a deoth of one foot. 

It was felt that some form of transverse leveling was 
necessary. Adjustment bolts were an obvious solution but 
the location was yet to be selected. The first sketches in- 
corporated adjustment bolts between the channel and the bot- 
tom plate. This produced an indeterminate situation with 
regard to flexure, inconvenient locations for adju-tment, and 
high fabrication cost. The subsequent designs placed the 
adjustment bolts between the channels and the main beams 
which gave the botcom plate full support across its width at 
2 foot intervals. 

The bottom plate was designed slightly wider than the 
flume width. This permitted attaching an angle to hold the 
bottom edge of the vertical plate fixed. The upper edge 
of the vertical plate had nuts welded on at the two foot 
points. Studs were attached between the nuts and the 



14 

vertical angles to suo 'ort the plate and nrovide an ad- 
justment for its longitudinal alignment. The inside of the 
flume was finished with an epoxy resin apdied with a hand 
roller. The flume construction is shown in figures 4 and 5. 
( Appendix C) 

In operations of this kind, guide rails ar^ generally 
fixed in the level position. They then may be used to con- 
trol the slope. oince the rails were attached to the flume, 
direct slope measurements was not possible. instead, dif- 
ferences between the surface of a standing body of water and 
the flume floor were used to measure the slope and calibrate 
a revolution indicator which served as the primary method of 
slope control. 

Tailgate 
Control over the depth was exercised by a gate mounted 
at the end of the flume. The gate was manually operated 
from the side of the flume. Figure 6 (apoendix C) shows 
the gate. The gate was made in such a way that it could 
be used either as a sluice or as a weir. Throughout this 
first series of tests, the gate was used exclusively as a 
weir. 

Forebay 
The forebay, 8 feet wide and 10 feet long, was con- 
structed of plywood and lined with sheet metal, and is shown 
in figure 10 of apoendix C. The 3 inch and 6 inch pipes 
entered the rear of the forebay at the top. The 6 inch 



15 

line was centered and the 3 inch liiEwas placed slightly off 
center. The diffusing mechanism for each supply line con- 
sisted of a tee and cross pipe of the sane size as the line 
at the bottom of the forebay. The turbulence of the entering 
water was controlled by a 4 inch gravel baffle and three wire 
mesh screens of 13 mesh per inch. The transition section con- 
tinuing into the bottom and side walls of the flume was made 
of quarter ellipses in the horizontal and vertical planes 
respectively with a ratio of major to minor axes of 1.5 to 
1.0. The joint between the flume and the forebay was sealed 
with a flexible rubber gasket mounted so as not to interfere 
with the flow. 

Slooe Control Mechanism 
The flume rests upon six screw jacks and a hinge. The 
hinge is located at the joint of the flume and the forebay. 
The jacks are similar in all respects with the exception of 
the gear ratio. The jacks are divided into three pairs with 
rates of raise of one, two, and three inches for 96 turns of 
the shaft. aince the hinge was a fixad point and the opposite 
end of the flums was the point of maximum movement, the jacks 
were arranged such that the pair nearest the hinge moved the 
least and the pair at the opposite end of the fluma had lar- 
gest displac 3mant per revolution. This maintained the bottom 
of the flume as a plan? while it was baing raised and lowered. 



The jacks on each side of the flume were driven by a 
common 1 inch shaft line connected at one and to a 90 miter 
gear. The miter gears on ei^har side in turn were connected 



16 

to a single 60:1 ratio gear reducer. The power to operate 
all the jacks was supplied by a 1-1/2 horsepower, 1750 re- 
volutions per minute, reversible, electric motor connected 
directly to the gear reducer. This provides a rate of ver- 
tical displacement at the dov/nstream .lack station of ap- 
proximately 1 inch per minute. 

The jacks were arranged in such a way that the downstream 
end of the flume may move from 12 inches below hori .ontal to 
3 inches above horizontal, resulting in a maximum positive 
slope of 1/60 and a maximum adverse slope of 1/240. The 
motor was controlled by a raise, lower, and stop control 
switch. Safety switches were located both near the motor and 
near the control switch. It was necessary to unlock these 
before tne control circuit could be completed. In addition, 
automatic limit switches were provided to prevent running the 
jacks beyond their limits. The general arrangement of jacks 
and gears is shown in figure 7» 

In order to connect the ends of the jacks (which move 
in a vertical line) to the flume (which moves in an arc), it 
was necessary to use a pinned linkage. A photograph of the 
linkage is shown in figure 8. (Appendix G) 

Due to the arc of the linkage, the relation between the 
rise of the flume and the revolutions turned by the jack 
shafts was not linear. Therefore, it was necessary to make • 
a calibration of bhe slope rather than computing it. 

At the time of erection, the jacks were leveled at .001 
foot before the flume was erected. During the alignment oro- 
cedure the ja^ks were raised or lowered individually as needed 



17 

to obtain a level base. The shaft couplings were then in- 
stalled an no further individual movements were made. 

Construction 
Considerable time was spent in obtaining the requi- 
sitions, bids from several contractors and actual super- 
vision of the erection of the flume. The foundations, 
jack piers, plumbing, and electrical controls were installed 
by Purdue university Physical Plant. The structural parts 
of the flume were built and assembled by a contractor, 
the flume adjustments, construction of the instrument 
carriage, installation of manometers and calibrations were 
done by the Research Assistant and student labor »vhen needed, 



18 



PREPARATION FOR TESTING 

Slope Calibration 

After the flume was aligned and leveled a slope cal- 
ibration was made by visually counting the revolutions of 
the slowest speed shaft and measuring the depth of a still 
■dooI of water at two points 50 feet apart. A steel tape was 
installed with station at the upstream end and station 
64 at the downstream end of the flume. The points chosen for 
slope measurement were stations 6 and 56. These points had 
previously given consistently good results when measurements 
were made of the distance betwe n the rails ani the flume 
floor. The calibration of slope versus revolutions appears 
in figure 9« (Appendix 0) The apparent scatter of the points 
toward the downstream end of the flume is due to the magnifi- 
cation resulting from the logarithmic scale at that end. 

In order to avoid the necessity of visually counting 
shaft revolutions to keep track of the sllope, a revolution 
indicator was made and installed at jack station number 3« 
The lowering of the flume activated the pointer which both 
multiplied and reversed the motion. The tip of the t>ointer 
rode on a lucite strip mounted below the motor controls. A 
mark was scribed on the lucite strip at each revolution over 
the range through 40 as well as at the test slopes. On 
the end of the shaft a circular lucite plate divided into ten 



19 

parts was mounted along with a fixed, pointer. Dhe slope 
desired was set by using the large indicator to the nearest 
revolution and setting the tenths of a rsvolution by using 
the small dial. This equipment was liter replaced by a com- 
mercial revolution counter mounted at the same location. 
This counter read directly to a tenth of a revolution and 
the reading could be interpolated to one half of that. This 
device provides a slope control with an accuracy of + 0.0000025 
feet/feet. Figure 8 (Appendix G shows the counter) 

vVater Supply and Measurement System 
The water available in the laboratory is recirculated 
through the system by two pumps rated at 300 Gpm and 2000 Gpm. 
The 3 inch discharge line from the 300 Gpm pump was connected 
to a new 3 inch line. This line contained a new 3 x 2,25 
inch venturi accurate to 0.5% over the range from 30 Gpm 
to 300 Gpm. The line was installed using long sweep elbows 
to reduce head loss. A 60 inch differential manometer reading 
to 0.01 inch was connected to the venturi and filled with 
tetrabromoethane (Specific gravity 2.95) which gave a man- 
ometer deflection of 51 inches with a flow of 336 Gpm. 

The 2000 Gpm pumo was connected to an existing 6 inch 
line which was improved by the installation of long sweep 
elbows in place of tee's and short elbows. In adlition, a 
new 6 x 4.176 inch venturi accurate to 0.5% over the range 
200 Gpm to 2000 Gpm was installed preceeded by a set of 
straightening vanes. A 30 inch differential manometer reading 



20 

to 0.01 inch was connected to the venturi and filled with 
mercury (Specific gravity 13.6). This manometer gave a de- 
flection of 14.9 inches for a maximum flow of 1790 Gpm. 
In both cases the Venturis were fitted with air vents to 
insure proper measurements. After a portion of the tasts 
had been made and the data evaluated, it was deemed necessary 
to improve the discharge measurements. The 60 inch manometer 
was attached to the 6 inch venturi and recalibrated using tet- 
rabromoethane as the manometer fluid. This resulted in a 
larger manometric deflection improving the accuracy of the 
measurement. The 50 inch manometer was connected to the 
3 inch venturi but thare was no need to recalibrate the meter. 

In order that the calibration of the venturi meters should 
have no error larger than that of the venturi meter, the scale 
to be used for the calibration was checked against standard 
weights by the indiana State Board of Health, Division of 
.'."eights and Measures. The scale error was less tnan 0.2-b 
or 2 pounds per 1000 pounds. For the purpose of calibrating 
of the venturi meters, branch lines led to a baffled concrete 
channel located above the weighing tank. 

At the point immediately before the 3 inch ani 6 inch 
lines entered the forebay, valves and valve bypasses were 
installed. The 6 inch line had a 2 inch bv-pass and valve 
and the 3 inch line was fitted with a 1 inch bypass and 
valve. The manometers were mounted in a position easily 
visible to the person adjusting the valves. The overall 
apparatus arrangement in the laboratory is shown in figure 
10. (Appendix G) 



21 



Venturi Calibration 
As soon as the essential piping was completed, cal- 
ibration of the venturi meters was begun. The procedure was 
as follows. The flow was set using a valve located downstream 
of the venturi and a waiting period of aporoximately 5 minutes 
was allowed foe the system to come to equilibrium under the 
new flow. ihe scales were preset to an arbitrary weight and 
the weighing tank valve closed. The manometer deflection was 
noted ani the water diverted into the weighing tank. The 
scales wer? tripoed and the timer manually started 7/hen the 
weight of water collected equalled the weight which had been 
preset on tne scale beam. Tne scale weight was noted. This 
method of calibration avoids the errors of non instantaneous 
starting and stopping of flow but still does not correct or 
make allowances for the difference in tne impact force of the 
water entering an empty tank as compared to a partially full 
tank. This er?or is of the magnitude of 1% which is com- 
patable with accuracy of the remainder of the system. fhe 
intervals oi' calibration were selected so as to fall between 
one and two inches of deflection on the 60 inch manometer 
containing the lighter fluid and not to exceed 1 inch on the 
30 inch manometer containing the mercury. The calibration 
curves are presented in figures 11, 12 and 13- (Appendix J) 



22 



MODEL TESTING 

model Construction 

From results of the preliminary experiments, (figure 
14- Appendix C) it was found that the predominant variable 
was the contraction ratio and that the length of the model 
had little or no influence for Froude numbers less than 0.5. 
This range of Froude numbers corresponds to the case usually 
found in practice. It was therefore decided that the first 
series of tests would concentrate on two dimensional sharp 
edge semi-circular models with no skew. 

The cost of machining mild steel plates to produce the 
constrictions was prohibitive. An estimate was obtained of 
8125.00 per model. The frequency of handling and changing 
the constriction plates also necessitated a model that wis 
light weight and still capable of sustaining hard use without 
damage. The models, as finally made, consisted of back up 
sheets of 1/2 inch exterior grade plywood faced v/ith a sheet 
of 22 gauge galvanized iron and braced by a steel angle. The 
openings were cut out of the galvanized iron sheet accurate 
to 1/32 inch. The ooening in the plywood backing had a radius 
of 1/2 inch greater than that used in the metal. The metal 
was bolted tightly against the wood. In the flume, the model 
was positioned with the metal face upstream. The construction 
of the models is shown in figure 15. (Appendix C) 



23 

Four models were made with contraction, ratios of 0.3, 
0.5, 0.7, and 0.9. At the slopes and flows tested, the 
model with m = 0.3 was submerged on a majority of the 
tests. The data oresented in Appendix G is that collected 
for the other three models which are shown in figure 5- 
(Aopendix C) 

Free Surface Measurement 
The position of the free surface was measured with an 
electric indicating point gaga reading to the nearest 0.1 mm. 
The point gage was mounted on an aluminum and brass bar in 
such a way that the gage could traverse the width of the flume 
This bar in turn was part of a carriage which rolled on the 
stainless steel rails along the flume. The carriage was 
rectangular and rolled on 4 wheels, two of which on one side 
of the carriage were grooved to provide alignment. The 
power supoly was mounted at the back of the carriage, The 
operator rode on a second carriage which was on its own rails. 
Details concerning the instrument carriage can be seen in 
figure 16. (Appendix C) 

The head of the point gage was com prised of two probes 
approximately 1/4 inc:h in diameter. The probes were sep- 
arated a distance of about 1-1/2 inches. The rear electrode 
was 15 millimeters longer than the front, and served as the 
ground. The end of the front electrode was pointed and was 
adjusted to the water surface. in operation, the obstruction 



to the flow presented by the rear electrode caused a rise in 
the water surface of v /2g against the electrode. The effect 
of this disturbance extended upstream to the measuring probe 



24 

and made it impossible to measure the true water surface. 
This effect was not a constant since the velocity head was 
different in each case. 

In order to improve the performance of the instrument, 
the rear electrode was replaced by a small diameter copper 
wire. This wire presented a much smaller obstruction to the 
flow and the operation of the gage was not only simplified, 
but gave data that should be superior to that obtained with 
the former arrangement. 

Uniform Flow Calibration 
Preliminary tests were run to determine uniform flow 
conditions in the flume. The variables involved are: the 
discharge, the slope and the tailgate settings. For each flow 
and slope the tailgate setting was determined by trial and 
error until uniform depth was obtained along the largest pos- 
sible reach of the flume. The models were located in such a 
way that the regain curve was complete within the uniform 
depth section, effects of boundary layer growth were not 
considered, and the flow was considered uniform as long as 
the depth remained constant. 

As the model tests progressed the uniform flow con- 
ditions were recorded in the auxiliary graphs of figures 
20 and 21. (Appendix J) 

The first was the normal depth versus the slope and 
the second was the height of the tailgate versus the slope. 
In both cases, the rate of flow was the parameter. In order 
to obtain data that would provide a complete coverage of the 



25 



range of Froude numbers investigated, the Froude number was 
first chosen and the normal depth computed in the following 
manner. 



f c 



v a 



or 



tt 



3 . Q* 



/T 2 3 Vf 



Gf x '/3 



from which 

" [Fo* B VJ 

After the desired normal iepth was computed, the slope 
for a particular flow which would give this value could be 
determined from figure 20 (Appendix 0). Then, entering fig- 
ure 21 (Appendix G) with the slope and flow, the necessary 
gate setting to produce uniform flow could be determined. 

Testing 
After the slope and uniform flow calibrations were com- 
pleted the testing was begun. The procedure used was to set 
the flow, which was the most difficult quantity to adjust, and 
let it remain constant while the slope, tail gate setting 
and models were changed. For each case of slope an I flow, 
the tail gate heighth was set according to the previous un- 
iform flow calibrations. In the majority of cases the con- 
ditions of uniform flow was obtained on the first trial and 
in every case no more than two trial gate settings were 
needed. Once the system had come to equilibrium and the 



26 



normal depth was obtained, a model was installed in the 
flume. For the first few runs, the complete profile was 
taken. It was observed that in the latter oortion of the 
regain curve, the water surface fluctuated vertically as 
much as one centimeter from one minute to the next. This 
fluctuation did not occur rabidly but rather slowly. Testing 
was immediately suspended while the cause for this was deter- 
mined. The first oossible cause investigated was tnat of a 
variable inflow into the flume. However, observation of the 
backwater showed it to be very stable. This indicated that 
the flow into the flume was constant and the phenomena was 
due to the model or that portion of the flume beyond the model. 
Eddies were suspected as a possible cause of this phenomenon. 
In an effort to eliminate eddies, the gap between the metal 
facing and the plywood back up board was filled and beveled 
to oroduce an angle of 4-5 . This resulted in no appreciable 
change. The condition most suspect however, was channelization 
of the flow to one side of the flume with eddies on the opposite 
side. It seemed that an instability was developed by the 
constriction. The reason for this channelization of the flow 
is not known. Misalignment of the bottom was not a factor 
because it v/as level within 1 millimeter throughout the length 
of the flume. The tail gate was leveled to an accuracy of 
1 millimeter an i the models were installed, using a square 
to check the alignment both horizontally and vertically. It 
was discovered that in a given case with most of the flow on 



27 



one side, after artificially deflecting the flow to the other 
side, it would remain on that side. This suggested that the 
lack of symmetry was not a factor. One possible solution is 
the addition of roughness and increasing of slope to sta- 
bilize the flow. However, since these tests were to be 
specifically smooth boundary tests, this was not done and 
the affected portion of the regain section was neglected. For 
the remainder of the tests, only a short profile before and 
after the model was taken to locate the points of maximum 
and minimum depth. Figures 17, 18, and 19 (Appendix 3) 
are photographs of the flume with model in place. Figure 
18 shows the flow going to one side and figure 19 shows the 
flow centered. 

Test Results 

The test data and the calculated values of the dis- 
charge coefficient CL., of the friction coefficient f, and 
of the Reynolds number are presented in table I. The ratio 
of the backwater depth to th3 normal depth y, /y is plotted 
versus the ratio of tne velocity head to normal depth with 
the contraction ratio m as a parameter in figure 22. The 
ratio of the velocity head to the normal depth is a measure 
of the kineticity of the flow, it is exactly half of the kin- 
eticity as defined by Bakhmetef f v yj , or half the square of 
the Froude number. The discharge coefficient C~ calculated 
from equation 2 of Appendix B is plotted versus the ratio of 
tne velocity heal to the normal depth with tne contraction 
ratio m as a parameter in figure 23. 



28 

The consistency of the data is well illustrated by the 
lack of scatter of the experimental points as plotted in fig- 
ures 22 and 23. (Appendix C) These test results may be com- 
pared with the small scale tests for the contraction ratio 
of 0.5 which is common to both test series. Inspection of 
figure 14- (Appendix 0) and figures 22 and 23 (Appendix G) 
show that the values are almost identical. For example, at 
a Froude number of 0.2, the value of the discharge coef- 
ficient Cj. from the small scale tests (figure 14- Appendix 
C) is 0.38 and the superelevation ratio y, /y is 1.1. 
Compared to this, the large scale tests, (Figure 22 and 23 
Appendix C) give G~ as 0.275 and 7-,/y as 1.119 for a kin- 
eticity of 0.02 which corresponds to the Froude number of 
0.2. At a Froude number of 0.4- the results of the small 
scale tests indicated G„ was equal to 0.53 and y, /y was 
1.4. Similarly, at a corresponding kineticy of 0.08, the 
large scale tests showed G n to be 0.4-83 and y, /y to be 
1.4-32. The reliability of the data may be better discussed 
in terms of the values of the friction coefficient and of 
the backwater ratio. 

The Darcy-Weisbach friction factor and the Reynolds 
number for the uniform flow established before each test 
were calculated in table I as follows: 

v 

r = 89 RS 



29 



The experimental friction factors were compared to the 
theoretical values obtained by adapting the Blasius and 
Prandtl formulas for flow in smooth pipes to the rectangular 
open channel. 

Th9 formulas for smooth pipe flow are: 

Blasius f = 0.3/6^ (W) -% R</0 S 

Prandtl-Von Karman _J_ - £ Q /oo fVJ? \Tf) -O.8 
Replacing D by 4R and simplifying, the equations become: 
Blasius f~- O.Z23 (^) V * 

Prandtl-Von Karman -J^ =Z.O /oa ( VR ]ff ) +OAO 

(12) '^ V 

Powell and Posey v J , working with a triangular flume 

found the formula governing their friction factor to be 

/ = 2.074- /o 3 (/RfT) -0-797 for 

tranquil flow in a smooth channel. The comparison between 
experimental values and the theoretical formulas is shown 
in figures 24 and 25 • (Appendix G) In figure 24 the ex- 
perimental values of the Darcy-Weisbach coefficient f are 
plotted versus the Reynolds number, both f and fR are 
as defined above based on the hydraulic radius. Also 
plotted on the same figure are the Blasius and Von-Karman 
relationship as well as the general range of experimental 
values obtained by Lansford and Robert son v ' J for smooth 
triangular channels. 

In figure 25 (Appendix C) the friction coefficient f 
and the Reynolds number are basad on the normal dapth, 
assuming two dimensional flow. Although this assumption 
is not completely true for the depth to width ratio used in 



50 



the experiments this was done to comoare the data with the 

(140 
experiments of Owen v ' which were done in a glass channel. 

In general, as shown in figure 24, (Aooeniix 0) the data 
fall above the theoretical lines which are a lower limit 
for a perfectly smooth boundary. The average friction co- 
efficient is f =0.021 which corresponds to an absolute 
roughness of approximately £ = f (f) = 0.0025 feet. This 
corresponds to the irregularities of the epoxy resin of the 
channel finish. 

The percentage of probable er ^or of the friction co- 
efficient is calculated as follows. The calculations are 
made based on a flow of 5 cfs ani a slope of 0.000100 foot/feet, 
The measurement of the flow during calibration had a possible 
er :or of 1% due to neglecting the impact in the weighing 
tank with different water depths, and the scales had a pos- 
sible error of 0.2%. The possible error in the Venturis 
was 0.5% and the observed manometer readings could have 
been in error by 0.02 inches. With the given flow, the 
manometer deflection was 52 inches of tetrabromoethane . 
This means that the possible error in reading the deflection 
value was approximately 0.02/54- or 5.85%. Since the flow 
varies with the 0.5 power of the deflection, this would 
represent an error in the flow of 1.95%. Therefore, the 
total possible er^or in the flow is 5.65%. The slope was 
calibrated and set to less than one tenth revolution of the 
connecting shafts. A maximum error of one twentieth of a 
revolution is equal to 0.0000025 feet/feet. At the slope 



31 

chosen, this would produce an error of 0.0000025/. 0001 or 
2.5%« The measurement of y was made to 0.1 millimeter. 
An error of this magnitude with the value of y found for 
this condition (21.70 cm) amounts to 0.01 cm/21.70 cm = 0.046%. 
The error in computing the wetted perimeter could be 

2(0.01 + 0.5)cm/192 cm = 0.053%. Since f = 8gRS/\/ 2 or 

3 2 
8gSy /Q p, the error in f can be expressed as 

V 2.5 2 + (3 x 0.046) 2 + (2 x 3.63) 2 + (0.053) 2 = '58.97 = 

7.68. 

At flatter slooes or lower flows, this error would be 
even larger than calculated. Oonversly, those tests at 
steeper slopes and higher flows should give the most nearly 
correct values of f. however, the points T^ould not be ex- 
pected to fall on the theoretical line since the materials 
used in the construction of the flume and the finish applied 
certainly caused some finite value of roughness. 

A. second check was made by determining the backwater 

ratio h,/4 h for each model test and comoaring the values 

( 6") 

obtained to those presented by Tracy and Jarter v } for 

rectangular constrictions. This comparison is shown in 
figure 26 (Appendix C) which shows that the backwater ratios 
are of the same order of magnitude although different as 
may be expected with different boundary geometries. 



32 



SUMMARY OF RESULTS /LNU CONCLUSIONS 
Flume Design 
The flume, as designed and built has proved adequate 
to carry out the proposed experiments. The entire apparatus 
necessary to carry out th? testing program can be run con- 
veniently by two men. The slope controls, including the 
jacks ani motor, function well and changing slope takes less 
than 5 minutes. 

Testing 
Only the first series of tests, sharp edged, semi- 
circular constrictions, have been completed on a large scale 
so comparisons cannot be drawn as such. However, several 
things indicate the validity of the lata. 

1. Close agreement with the small scale tests. 

2. Comparison with other investigators on: 

a. Friction factor. 

b. Mannings roughness 

3. Adherance of the plotted lata to a well 

defined pattern with little experimental 

scatter. 
The conclusions possible so far are primarilly drawn 
from figure 23 which relates the lischarge coefficient and 
the kineticity of the flow, and figure22 which shows the 
backwater in terms of kineticity. From the first, it can 



33 

be seen that above a value that corresponds with a Fronde 
number of 0.5, the Cj, value ceases to depend on the kin- 
eticity of the flow. The second, figure 22 shows the de- 
pendance of trie backwater value on both the contraction ratio 
and the kineticity. 



34 



BIBLIOGRAPHY 

1. ilusain, 3. T., "ir'reliminary WiOdel Investigations of 

Hydraulic Characteristics of River Flow Under Arch 
Bridges" Masters Thesis, Purdue University, 1959. 

2. Owen, H. J.; Sooky, A; Husain, S. T.; Delleur, J. W. ; 

" Hydraulics of River Flow Under Arch Bridges - A 
Progress Report." Progress Report submitted to the 
Board of the Joint Highway Research Project, May 14, 
1959. 

5. Lane, E. W. , "experiments on the Flow of Water Through 
Contractions in an Open Channel ' Transactions ASCE 
Vol. 83, 1919-1920. 

4. Kindsvater 0. E.; Garter, R. W. ; "Tranquil Flow 

Through Open Channel Constrictions". Transactions 
ASCe, Vol. 120, 1955- 

5. Tracy, H. J.; Carter, R. ft'.; "Backwater Effects of 

Open Channel Jonstrictions" . Transactions, ASCE, 
vol. 120, 1955. 

6. vallentine, n. R. , "Flow in Rectangular Channels 

with Lateral Constriction Plates". La Houille Blanche, 
January, 1958. 

7. Magler, F. A.; "Obstructions of Bridge Piers to the 

Flow of water". Transactions ASCE, Vol. 82, 1918 
PP354-95- 

8. Bakhmeteff, B. A.; "Hydraulics of Open Channels". 

Engineering Societies monograph, McGraw-Hill 
Series, 1932. 

9. U.S. Bureau of Public Roads, "Computation of Back- 

water Caused by Bridges". October, 1958. 

10. Albertson, m. L. ; Dai, Y. B.; Jensen, R. A.; Souse, 

H.; "Diffusion of Submerged Jets" Transactions, 
ASCE, Vol. 115, 1950. 

11. Henry, H. R.; "Discussion of 10 " Transactions, 

ASCE, Vol. 115, 1950. 



35 



12. Powell, R. W.; .Posey, C. J.; "Resistance Experimants 

in a Triangular Channel". Journal of the Hydraulics 
Division, Proceedings ASGE, May 1959. 

13. Lansford >v. ivi. ; Robertson J, M. Discussion, trans- 

actions Ao"E, Vol. 123, 1958 p. 707. 

14. Owen, W. to. ; "Laminar to Turbulent Flow in .Vide 

Open Channel" Transactions ASCE, Vol. 119, 1954. 



APPJSADIX A 
NOTATIONS 



36 



NUTATIONS 
SYMBOL UimIT DEFINITION 



2 
A feet Area of flow. 

B feet Stream width at bridge 

site or flume width. 

b feet Width of constriction 

opening equal to di- 
ameter of semi-circle. 

G-pj Coefficient of discaarge 

equation. 

D feet Pipe diameter. 

E An infinite series of 

powers of y /r. 
t o 

F Froude number of flow. 

Subscript o refers to 
unconstricted stream. 

f Darcy-Weisbach friction 

factor. 

g feet/sec Acceleration of gravity. 

h, feet Superelevation of back- 

water above normal 
depth. 

h^ feet Boundary friction loss. 

A k feet or Elevation difference be- 

centimeters tween points of maximum 



37 



Lr 
M 



cf s 



feet 



feet 



feet 



feet 



cfs 



cfs 



and minimum depth. 

Backwater head loss co- 
efficient . 

Conveyance of a channel 
or section of channel 
= AG \fR. 

Length of reach in back- 
water computations. 

Accumulated length in 
backwater computation. 

Scale of length. 

Channel contraction 
ratio equal to 1-b/B 
(in review of lit- 
erature) . 

Channel contraction ratio 
equal to b/B. 

Manning roughness or 
hydraulic exponent in 
K = const, y 

Wetted perimeter of stream 
or subsection. 

Total flow (subscripts 
m and p refer to 
model and prototype). 

flow in a subsection 
(subscripts m and p 



38 

refer to small model 
flume and to prototype 
flume. ) 

R feet Hydraulic radius. 

r feet Radius of semi-circular 

constriction. 

nn Reynolds number. 

S feet/feet Slope of stream bed or 

flume. 

V feet/sec Average velocity (sub- 

script o refers to un- 
constricted flow). 

y feet or Depth of flow. 

centimeters 

y feet or Normal deoth of flow in 

o 

centimeters unconstricted channel, 

y-, feet or Maximum depth of flow 

centimeters upstream of constriction. 

c<_ Slope of line on Con- 

veyance versus Depth 
graph. 
P Ratio of bottom slope to 

critical slope. 

7f ' y/y o 

Bakhmeteff backwater 

function. 



APPENDIX B 
EQUATIONS Oi' FLOW 



39 



UJiRIVATi-OlMb OJB' i^UATIOrJS GOVERNING THE PLOW IN RECTANGULAR 
CHANNELS WlTn fc>EMl-0±RCULAR CONSTRICTIONS 
The equation for the discharge in rectangular channels 
with a sharp crested serai-circular constriction is ob- 
tained and is expressed in terms of an infinite series of 
powers of the ratio y-,/r. With reference to Figure 27, 
(Appendix C) Bernoulli theorem gives 

V= CV23V - C V2 9M-h) 

The element of area is 

d A-- 2 Vr^TT 2 dh 

and the discharge is thus 

Q '/VdA *{*'CVZjttrh)-2h*-h* dh (i) 

Expanding into a series, integrating term by term, and making 
"use of the fact that 2r = b: 

Q-C D vgy g y* b[)-0.IW($f-QM&)$~3 '(2) 

This may be written as 

Q -- C ^ fc £ (?) 

where 
and 

E - {[-O.IZ?4(^f~ O.OI77(^)* -t ■■] (5) 

The discharge in the rectangular flume is given by 

q = VoAo = Fo *f & y* /z (6) 



where 

(7) 






40 



is the i'Toude number of the undisturbed flow. Equating 
(.2; and (6) and solving for the coefficient of discharge 

Co -^-A (%■)% 

where 

m = 4 (9 ) 

or 

5o L /7 /77^"Cp J 



APPENDIX C 
FIGURES AMD TABLES 



41 



-SECTION 


1 SECTION 


2 


SECTION 3- 


) t 


W//\ 

KLJ / / / 


V/f 


^< 


W//ffi 


3' 


S^ 


{ js 




^lV 


f- 




h- 30- 




I 





UPSTREAM OF BRIDGE 



30'- 



THR0U6H BRIDGE 



IDEALIZED STREAM CROSS SECTION 



FIGURE I 



42 





o 
























o 

O 

O 

_] 
























CONVEYANCE vs. DEPTH 

FOR BRIDGE S79 
(CLAY COUNTY, INDIANA) 




















































































1 


\ 
























\ 
























\ 






















\ 


\ 






















\ 




1 




\ 


L 


in 




























\ 


0) 
























\ 


























\ 


























> 
\ 


























\ 




























\ 


























\l 
























\ 


\ 


























\ 


























\ 






o o 

ro — 

>i 901 


■o 



CM 
UJ 

q: 

CD 



43 




TO 
UJ 

tr 

(3 



44 




45 




FIGURE 5 



FLUME AND MODELS 




FIGURE 6 
TAIL GATE CONSTRUCTION 



46 




3SIVH .1 SO 



oj «ao» jo SNuru 96 ry/ 
#13S »ovr ^ v Tt j 



S 



ft 



f-f©1- asi»a „2 aoj »bo» , 
Lr-^]^ 2#13S X 




r2 



£ 



3SIVS ,.C HOJ WHOM JO SNUHJ. 96 

£ * 13S MDVr 




CO 

o 
< 



Ll 

O 
L±J 



47 




FIGURE 8 



JACK DETAIL 



48 

































^ 


( 


S> 




o 

~o 

o 

2 1 
o u. 






























SLOPE CALIBRATIO 

CURVE FOR 
HYDRAULIC TESTIh 
FLUME 






















































































































































































































































































































































































( 


\ 


























































1 

.000010 oc 

SLOPE- 


















































































































































\ < 


i 


























































































































< 


!\ 














































t 


> 


































































































































































































































































































































O O ffi id S HI " f 

o — 

SNonnnoA3a 





49 




CD 

< 

cr 

CT 
< 

CO 

Z) 

< 
cr 

a. 
< 



V 



U2 



50 



.a 






























" > 


u 


.5 






























/ 
































/ 




.5 




























j 


f 






























/ 






5 


























J 


/ 






























/ 








5 


























/ 






























/ 


































/ 
































/ 












































































































































































































CAL 


BRAT 
REE- 


ION C 
INCH 


URVE 
VENT 


FOR 
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